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Theorem eusv2nf 4134
Description: Two ways to express single-valuedness of a class expression A(x). (Contributed by Mario Carneiro, 18-Nov-2016.)
Hypothesis
Ref Expression
eusv2.1 A V
Assertion
Ref Expression
eusv2nf (∃!yx y = AxA)
Distinct variable groups:   x,y   y,A
Allowed substitution hint:   A(x)

Proof of Theorem eusv2nf
StepHypRef Expression
1 nfeu1 1889 . . . 4 y∃!yx y = A
2 nfe1 1362 . . . . . . 7 xx y = A
32nfeu 1897 . . . . . 6 x∃!yx y = A
4 eusv2.1 . . . . . . . . 9 A V
54isseti 2537 . . . . . . . 8 y y = A
6 19.8a 1460 . . . . . . . . 9 (y = Ax y = A)
76ancri 307 . . . . . . . 8 (y = A → (x y = A y = A))
85, 7eximii 1471 . . . . . . 7 y(x y = A y = A)
9 eupick 1957 . . . . . . 7 ((∃!yx y = A y(x y = A y = A)) → (x y = Ay = A))
108, 9mpan2 403 . . . . . 6 (∃!yx y = A → (x y = Ay = A))
113, 10alrimi 1392 . . . . 5 (∃!yx y = Ax(x y = Ay = A))
12 nf3 1537 . . . . 5 (Ⅎx y = Ax(x y = Ay = A))
1311, 12sylibr 137 . . . 4 (∃!yx y = A → Ⅎx y = A)
141, 13alrimi 1392 . . 3 (∃!yx y = Ayx y = A)
15 dfnfc2 3568 . . . 4 (x A V → (xAyx y = A))
1615, 4mpg 1316 . . 3 (xAyx y = A)
1714, 16sylibr 137 . 2 (∃!yx y = AxA)
18 eusvnfb 4132 . . . 4 (∃!yx y = A ↔ (xA A V))
194, 18mpbiran2 834 . . 3 (∃!yx y = AxA)
20 eusv2i 4133 . . 3 (∃!yx y = A∃!yx y = A)
2119, 20sylbir 125 . 2 (xA∃!yx y = A)
2217, 21impbii 117 1 (∃!yx y = AxA)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1224   = wceq 1226  wnf 1325  wex 1358   wcel 1370  ∃!weu 1878  wnfc 2143  Vcvv 2531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-rex 2286  df-v 2533  df-sbc 2738  df-csb 2826  df-un 2895  df-sn 3352  df-pr 3353  df-uni 3551
This theorem is referenced by:  eusv2  4135
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